Download presentation

Presentation is loading. Please wait.

Published byJayda Pridmore Modified over 3 years ago

1
Infinities 2 sequences and series

2
9:30 - 11:00 Geometric Sequences 11:30 - 13:00 Sequences, Infinity and ICT 14:00 - 15:30 Quadratic Sequences

3
Starter activity Can you make your calculator display the following sequences? Find the 20 th term for each of these sequences.

4
Can you find the next two terms of the following sequence 4, 8, 16, 32,....? Geometric Sequences

5
Geometric sequences Position number 123456 Sequence48163264128 x2 4, 8, 16, 32,.... A sequence is geometric if where r is a constant called the common ratio x2

6
Geometric sequences or geometric progressions, hence the GP notation Different ways to describe this sequence: By listing its first few terms: 4, 8, 16, 32,... By specifying the first term and the common ratio: 1 st term is 4 and common ratio is 2 or By giving its nth term ? By graphical representation ?

7
Finding the nth term Position number 12345n Sequence48163264 4x14x24x44x84x16 4x2 0 4x2 1 4x2 2 4x2 3 4x2 4 nth term = 4x2 n-1

8
4, 8, 16,... is a divergent sequence

9
Geometric sequences Can you find the next two terms of the following sequence? 0.2, 0.02, 0.002,.... Can you describe this sequence in different ways? By listing its terms: By specifying the first term and the common ratio: By finding its nth term: By graphical representation:

10
0.2, 0.02, 0.002,... is a convergent sequence The sequence converges to a certain value (or a limit number)

11
e.g. it approaches 0 This convergent sequence also oscillates. Another example of a convergent sequence:

12
Geometric sequences 1. Can you generate (or find) the first 5 terms of the following GPs? Seq A: Seq B: 2. Can you write down the nth term of these sequences? 3. Are these sequences convergent or divergent? Can you use the limit notation in your answers?

13
Geometric sequences 1. What is the ratio and the 7 th term for each of the following GPs? Seq A: 2, 10, 50, 250,...? Seq B: 24,12, 6, 3,....? Seq C: -27, 9, -3, 1,....? Challenge 1 What if you want to find the 50 th term of each of these sequences? How would you change your approach? Challenge 2 The 3 rd term in a geometric sequence is 36 and the 6 th term is 972. What is the value of the 1 st term and the common ratio? Challenge 3 Q6 handout

14
Suppose we have a 2 metre length of string...... which we cut in half We leave one half alone and cut the 2 nd in half again... and again cut the last piece in half Geometric Series

15
Continuing to cut the end piece in half, we would have in total In theory, we could continue for ever, but the total length would still be 2 metres, so This is an example of an infinite series.

16
or is the Greek capital letter S, used for Sum

17
Geometric series The sum of all the terms of a geometric sequence is called a geometric series. We can write the sum of the first n terms of a geometric series as: When n is large, how efficient is this method ? S n = a + ar + ar 2 + ar 3 + … + ar n –1 For example, the sum of the first 5 terms of the geometric series with first term 2 and common ratio 3 is: S 5 = 2 + (2 × 3) + (2 × 3 2 ) + (2 × 3 3 ) + (2 × 3 4 ) = 2 + 6 + 18 + 54 + 162 = 242

18
The sum of a geometric series Start by writing the sum of the first n terms of a general geometric series with first term a and common ratio r as: Multiplying both sides by r gives: S n = a + ar + ar 2 + ar 3 + … + ar n –1 rS n = ar + ar 2 + ar 3 + … + ar n –1 + ar n Now if we subtract the first equation from the second we have: rS n – S n = ar n – a S n ( r – 1) = a ( r n – 1) Challenge: Can you follow the proof of the formula for the sum of the first n terms of a GS? (in pairs)

19
Geometric series a)Find the sum of the first 7 terms of the following GP: 4, - 2, 1,... giving your answer correct to 3 significant figures. b)Calculate: Challenge Is ? What is as an exact fraction?

Similar presentations

OK

THE CONCEPT OF SEQUENCE AND SERIES. Adaptif Hal : 2 THE PROGRESSIONS Hal.: 2 The Pattern of Sequence and Series Number Basic Competence: Applying the.

THE CONCEPT OF SEQUENCE AND SERIES. Adaptif Hal : 2 THE PROGRESSIONS Hal.: 2 The Pattern of Sequence and Series Number Basic Competence: Applying the.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on omission of articles in english Ppt on rbi reforms define Ppt on power line communication final year project Convert free pdf to ppt online convert Ppt on different types of computer softwares for resume Animated ppt on magnetism Free ppt on motivational stories Ppt on various types of web browsers and their features Ppt on honge oil Ppt on earth hour 2017